Solve frac{a+4n/23+2-sqrt{3}}{1+frac{6+9sqrt{3}}{23}*(2-sqrt{3})}

$$\frac { \frac { a + 4 n } { 23 } + 2 - \sqrt { 3 } } { 1 + \frac { 6 + 9 \sqrt { 3 } } { 23 } \times ( 2 - \sqrt { 3 } ) }$$

Factor

$$\frac{\left(3\sqrt{3}-2\right)\left(a+4n+46-23\sqrt{3}\right)}{92}$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2-\sqrt{3} times \frac{23}{23}.

$$factor(\frac{\frac{a+4n}{23}+\frac{23\left(2-\sqrt{3}\right)}{23}}{1+\frac{6+9\sqrt{3}}{23}\left(2-\sqrt{3}\right)})$$

Since \frac{a+4n}{23} and \frac{23\left(2-\sqrt{3}\right)}{23} have the same denominator, add them by adding their numerators.

$$factor(\frac{\frac{a+4n+23\left(2-\sqrt{3}\right)}{23}}{1+\frac{6+9\sqrt{3}}{23}\left(2-\sqrt{3}\right)})$$

Do the multiplications in a+4n+23\left(2-\sqrt{3}\right).

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{1+\frac{6+9\sqrt{3}}{23}\left(2-\sqrt{3}\right)})$$

Express \frac{6+9\sqrt{3}}{23}\left(2-\sqrt{3}\right) as a single fraction.

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{1+\frac{\left(6+9\sqrt{3}\right)\left(2-\sqrt{3}\right)}{23}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{23}{23}.

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{\frac{23}{23}+\frac{\left(6+9\sqrt{3}\right)\left(2-\sqrt{3}\right)}{23}})$$

Since \frac{23}{23} and \frac{\left(6+9\sqrt{3}\right)\left(2-\sqrt{3}\right)}{23} have the same denominator, add them by adding their numerators.

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{\frac{23+\left(6+9\sqrt{3}\right)\left(2-\sqrt{3}\right)}{23}})$$

Do the multiplications in 23+\left(6+9\sqrt{3}\right)\left(2-\sqrt{3}\right).

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{\frac{23+12-6\sqrt{3}+18\sqrt{3}-27}{23}})$$

Do the calculations in 23+12-6\sqrt{3}+18\sqrt{3}-27.

$$factor(\frac{\frac{a+4n+46-23\sqrt{3}}{23}}{\frac{8+12\sqrt{3}}{23}})$$

Divide \frac{a+4n+46-23\sqrt{3}}{23} by \frac{8+12\sqrt{3}}{23} by multiplying \frac{a+4n+46-23\sqrt{3}}{23} by the reciprocal of \frac{8+12\sqrt{3}}{23}.

$$factor(\frac{\left(a+4n+46-23\sqrt{3}\right)\times 23}{23\left(8+12\sqrt{3}\right)})$$

Cancel out 23 in both numerator and denominator.

$$factor(\frac{a+4n-23\sqrt{3}+46}{12\sqrt{3}+8})$$

Rationalize the denominator of \frac{a+4n-23\sqrt{3}+46}{12\sqrt{3}+8} by multiplying numerator and denominator by 12\sqrt{3}-8.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{\left(12\sqrt{3}+8\right)\left(12\sqrt{3}-8\right)})$$

Consider \left(12\sqrt{3}+8\right)\left(12\sqrt{3}-8\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{\left(12\sqrt{3}\right)^{2}-8^{2}})$$

Expand \left(12\sqrt{3}\right)^{2}.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{12^{2}\left(\sqrt{3}\right)^{2}-8^{2}})$$

Calculate 12 to the power of 2 and get 144.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{144\left(\sqrt{3}\right)^{2}-8^{2}})$$

The square of \sqrt{3} is 3.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{144\times 3-8^{2}})$$

Multiply 144 and 3 to get 432.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{432-8^{2}})$$

Calculate 8 to the power of 2 and get 64.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{432-64})$$

Subtract 64 from 432 to get 368.

$$factor(\frac{\left(a+4n-23\sqrt{3}+46\right)\left(12\sqrt{3}-8\right)}{368})$$

Apply the distributive property by multiplying each term of a+4n-23\sqrt{3}+46 by each term of 12\sqrt{3}-8.

$$factor(\frac{12a\sqrt{3}-8a+48\sqrt{3}n-32n-276\left(\sqrt{3}\right)^{2}+184\sqrt{3}+552\sqrt{3}-368}{368})$$

The square of \sqrt{3} is 3.

$$factor(\frac{12a\sqrt{3}-8a+48\sqrt{3}n-32n-276\times 3+184\sqrt{3}+552\sqrt{3}-368}{368})$$

Multiply -276 and 3 to get -828.

$$factor(\frac{12a\sqrt{3}-8a+48\sqrt{3}n-32n-828+184\sqrt{3}+552\sqrt{3}-368}{368})$$

Combine 184\sqrt{3} and 552\sqrt{3} to get 736\sqrt{3}.

$$factor(\frac{12a\sqrt{3}-8a+48\sqrt{3}n-32n-828+736\sqrt{3}-368}{368})$$

Subtract 368 from -828 to get -1196.

$$factor(\frac{12a\sqrt{3}-8a+48\sqrt{3}n-32n-1196+736\sqrt{3}}{368})$$

Consider 12a\times 3^{\frac{1}{2}}-8a+48\times 3^{\frac{1}{2}}n-32n-1196+736\times 3^{\frac{1}{2}}. Factor out 4.

$$4\left(3a\sqrt{3}-2a+12\sqrt{3}n-8n-299+184\sqrt{3}\right)$$

Rewrite the complete factored expression. Simplify.

$$\frac{3a\sqrt{3}-2a+12\sqrt{3}n-8n-299+184\sqrt{3}}{92}$$

Evaluate

$$\frac{\left(3\sqrt{3}-2\right)\left(a+4n+46-23\sqrt{3}\right)}{92}$$